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Homework Helper
2020 Award
Of course almost any set can be given some kind of group structure, but one which ahs nothing at all to do with its topological structure. E.g. any finite set can be made into a cyclic group and any countable set can be made into the integer group, and any set with cardinality equal to that of the reals can be made into the reals as a group.
infinitely small
What is the proof for the almost any set can be given some kind of group structure?Proofs for the others? Because i am interested in those proofs.Thank you.
mathwonk
Homework Helper
2020 Award
if S is a set of n elements, that means there is a bijection from S to the set Z/n = {0,1,2,....,n-1}. Since that set has a group structure as a cyclic group, namely integers modulo n, you define an isomorphic group structure on the set S by first fixing a bijection f:S-->Z/n, and then taking any two elements of S, you add them by going over to Z/n by the bijection f, adding them there, and then bringing their sum back by the inverse of the bijection f to get an element of S.
mathwonk
Homework Helper
2020 Award
Similarly any set that has a bijection to an group, cvan be made into a group in this way. Hnce any set with a bijection to the ral numbers, can be made into a group isomorphic to the additive goup of R. If this is puzzling, you would benefit from reading some basic set theory, about bijections and so on.